Enhanced Diffusion of a Needle in a Planar Array of Point Obstacles
The behavior of simple gases such as oxygen or helium is very well understood; ever since physicists Rudolf Clausius, James Clerk Maxwell and Ludwig Boltzmann developed the kinetic theory of gases at the end of the 19th century. But the situation for dense systems of very long, thin fibers - currently a field of highly intense investigation in nanotechnology - is completely different. When such a fiber veers, it immediately collides with many other fibers, and one observes strong mutual obstruction - much like in fluids.
But the scientists Dr. Felix Höfling, Professor Erwin Frey and Adj. Professor Dr. Thomas Franosch of the Arnold-Sommerfeld Center for Theoretical Physics at LMU Munich have now discovered a new, unexpected effect: despite the strong obstructions, diffusion of the fibers does not become slower; their speed dramatically increases instead. "From a simple model and extensive computer simulations, we have observed that the fibers move more than 100 times faster than expected," reports Franosch.
This effect can be attributed to a spacious zigzag motion of the fibers, where the paths they trace resemble the tracks of an ice skater. To put it in physics terms: one axiom of the kinetic theory of gases is the hypothesis of molecular chaos, which Boltzmann called the "Stoßzahlansatz". According to this, the motion of molecules after a collision is independent of their motion before the collision. The good thing about this is that it allows a statistical description using random processes, where the macroscopic properties are independent of the microscopic details of the collisions.
One result of the present work, however, is that this hypothesis is only partially correct for very long fibers. Although the motion is not deterministically predicable, it is not entirely random. Instead, the fibers travel for long distances in a straight line, changing their orientation only very slowly. As a result, microscopic details are amplified and become visible in the macroscopic zigzag motion.
The way a fiber is braked at the "peaks" of its zigzag motion and ultimately travels backwards explains the enhanced diffusion coefficient. Careful analysis reveals that the correlation between the diffusion coefficient and the density of the needles is a fractal. Although the scientists can now explain the fundamental mechanisms for enhanced diffusion, there are indications that the mathematics of fractals, chaos and quasi-periodic orbits will make up a large part of the work in developing a more precise description, which will in turn inspire more challenging fundamental research.
You can watch a simulation movie of this zigzag motion of needles on the Statistical Physics research field website: www.theorie.physik.uni-muenchen.de/lsfrey/research/fields/statistical_physics/2008_004/
"Enhanced Diffusion of a Needle in a Planar Array of Point Obstacles",
Felix Höfling, Erwin Frey, and Thomas Franosch,
Physical Review Letters, 19 September 2008
Adj. Prof. Dr. Thomas Franosch
Tel.: ++49 (0) 89 / 2180 - 4603
Fax: ++49 (0) 89 / 2180 - 4154